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May  2017, 37(5): 2813-2827. doi: 10.3934/dcds.2017121

Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system

Department of Applied Mathematics, National Chung Hsing University, 145, Xingda Road, South Dist, Taichung 402, Taiwan

Received  May 2016 Revised  December 2016 Published  February 2017

Fund Project: This work was partially supported by the Ministry of Science and Technology of the Republic of China under the grant 105-2115-M-005-002. The author would like to thank the referees for valuable comments.

We consider a three components lattice dynamical system which arises in the study of a three species competition model. It is assumed that two weaker species have different preferences of food and the third stronger competitor has both preferences of food. Under this assumption, it is well-known that there is the minimal speed such that a traveling wave solution exists for any speed above this minimal one. In this paper, we prove the monotonicity of wave profiles and the uniqueness (up to translations) of wave profiles for each given admissible speed under certain restrictions on parameters.

Citation: Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121
References:
[1]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. 

[2]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Diff. Eqns., 184 (2002), 549-569. 

[3]

X. Chen and J.-S. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. 

[4]

X. Chen, S. -C. Fu and J. -S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal. , 38 (2006), 233-258.

[5]

S. -N. Chow, Lattice dynamical systems, in J. W. Macki, P. Zecca (Eds. ), Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 1–102.

[6]

S.-N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. 

[7]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Lecture Notes in Biomathematics 28, Springer Verlag, 1979.

[8]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525. 

[9]

J.-S. GuoY. WangC.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829. 

[10]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346. 

[11]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. 

[12]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.

[13]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in: J. W. Macki, P. Zecca (Eds. ), Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 231–298 doi: 10.1007/978-3-540-45204-1_4.

[14]

E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511624094.

[15]

B. Shorrocks and I. R. Swingland, Living in a Patch Environment, Oxford University Press, New York, 1990.

show all references

References:
[1]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. 

[2]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Diff. Eqns., 184 (2002), 549-569. 

[3]

X. Chen and J.-S. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. 

[4]

X. Chen, S. -C. Fu and J. -S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal. , 38 (2006), 233-258.

[5]

S. -N. Chow, Lattice dynamical systems, in J. W. Macki, P. Zecca (Eds. ), Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 1–102.

[6]

S.-N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. 

[7]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Lecture Notes in Biomathematics 28, Springer Verlag, 1979.

[8]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525. 

[9]

J.-S. GuoY. WangC.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829. 

[10]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346. 

[11]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. 

[12]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.

[13]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in: J. W. Macki, P. Zecca (Eds. ), Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 231–298 doi: 10.1007/978-3-540-45204-1_4.

[14]

E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511624094.

[15]

B. Shorrocks and I. R. Swingland, Living in a Patch Environment, Oxford University Press, New York, 1990.

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